Learn Extracted exam questions AP Physics C: Mechanics 2022 Free Response · Set 2
2022 Free Response · Set 2
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A small sled slides across a rough horizontal table with an initial velocity $v_0$. The coefficient of kinetic friction between the sled and the table is $\mu_k$. A string connects the sled to a device on the ground. The device maintains constant tension $F_T$ in the string by unwinding the string as the sled slides to the right. The total mass of the sled is $m$. The string is attached to the device at $x = 0$ and at a height of $y$, as shown in Figure 1. The horizontal position of the sled is represented by $x$, as shown in Figure 2. Express all algebraic answers in terms of $m$, $\mu_k$, $F_T$, $x$, $y$, and physical constants, as appropriate.
[Figure 1: A sled on a horizontal table, moving right with initial velocity $v_0$. Positions $x = 0$ and $x = L$ are marked on the table top. Below the table, a string of vertical extent $y$ connects down to a "Device" on the ground.]
[Figure 2: Top-down/side view with axes $+y$ up and $+x$ to the right. The sled starts at the left, and a dashed line of horizontal length $x$ extends from the sled's starting position to its current position. A "String" is shown connecting the sled diagonally down to the Device on the ground.]
Note: Figures not drawn to scale.
On the dot below that represents the sled, draw and label the forces (not components) that are exerted on the sled a short time after $t = 0$ but before the sled has come to rest. Each force must be represented by a distinct arrow starting on, and pointing away from, the dot.
[A diagram showing a single dot with a horizontal dashed line through it and a vertical dashed line through it, crossing at the dot, for the student to draw force vectors on.]
Determine an expression for the angle $\theta$ that the string makes with the vertical when the sled has traveled a horizontal distance $x$.
Derive an expression for the normal force $F_N$ exerted on the sled by the table as a function of the position $x$.
Derive an expression for the magnitude of the net horizontal force $F_{net}$ exerted on the sled as a function of the position $x$.
Derive an expression for the work $W$ done by the string on the sled as the sled moves from $x = 0$ to $x = L$.
The sled comes to rest after traveling a horizontal distance $x = 2L$. As the system slides from $x = 0$ to $x = L$, the energy dissipated by friction is $E_1$. As the sled slides from $x = L$ to $x = 2L$, the energy dissipated by friction is $E_2$. Is $E_1$ greater than, less than, or equal to $E_2$ ?
___ $E_1 > E_2$ ___ $E_1 < E_2$ ___ $E_1 = E_2$
Justify your answer.
[Figure (top): A spring with spring constant $k$ is attached to a wall on the left, compressed by an amount $\Delta x$ from its natural length, holding Block 1 (mass $m_1$). Block 2 (mass $m_2$) sits at rest at position $x = 0$, to the right of Block 1, in front of a Motion Detector mounted on a wall at the far right.]
[Figure (bottom): After release, the spring is no longer compressed (uncompressed length shown), and the combined Block 1 + Block 2 system moves to the right with speed $v$, starting from $x = 0$.]
Block 1 of mass $m_1$ is held at rest while compressing an ideal spring an amount $\Delta x$. The spring constant of the spring is $k$. Block 2 has mass $m_2$, where $m_2 < m_1$. At time $t = 0$, Block 1 is released. At time $t_C$, the spring is no longer compressed and Block 1 immediately collides with and sticks to Block 2. The blocks stick together and the two-block system moves with constant speed $v$, as shown. Frictional effects are negligible.
The impulse on Block 1 from the spring during the time interval $0 < t < t_C$ is $J_S$. The impulse on Block 1 from Block 2 during the collision is $J_2$. Which of the following expressions correctly compares the magnitudes of $J_S$ and $J_2$ ?
___ $J_S > J_2$ ___ $J_S < J_2$ ___ $J_S = J_2$
Justify your answer.
On the following axes, draw graphs of the magnitude of the momentum of each block as a function of time, before and after $t_C$. The collision occurs in a negligible amount of time. The grid lines on each graph are drawn to the same scale.
[Two blank graphs labeled "Momentum" on the vertical axis and "Time" on the horizontal axis, each with a marked point $t_C$ on the time axis and dashed gridlines; left graph labeled "Block 1", right graph labeled "Block 2", for the student to sketch momentum vs. time before and after $t_C$.]
Show that the velocity $v$ of the two-block system after the collision is given by the equation $v = \dfrac{\sqrt{k m_1}}{m_1 + m_2}\Delta x$.
A group of students use the setup to perform an experiment. They measure the mass of Block 1 to be $m_1 = 0.500\text{ kg}$, and the spring constant $k$ of the spring to be $150\text{ N/m}$. The mass of Block 2 is unknown. They perform several trials and in each trial the spring is compressed a different distance $\Delta x$ and the final velocity $v$ of the two-block system is measured. They graph $v$ as a function of $\Delta x$, as shown below.
Draw a line that represents the best fit to the data points shown.
[Scatter plot: vertical axis $v$ (m/s) from 0.0 to 0.7 in increments of 0.1; horizontal axis $\Delta x$ (m) from 0.0 to 0.07 in increments of 0.01. Data points approximately at $(0.01, 0.08)$, $(0.02, 0.16)$, $(0.035, 0.28)$, $(0.05, 0.45)$, $(0.06, 0.58)$, $(0.065, 0.62)$, for the student to draw a best-fit line through.]
Use the best-fit line to calculate the mass of Block 2.
After the experiment, the students use a balance to measure the mass of Block 2 and find it to be greater than what was determined in part (d). To explain this discrepancy, one of the students proposes that the spring constant was incorrectly measured at the beginning of the experiment. The students measure the spring constant again and record a new value, $k'$.
Should the students expect that $k'$ be greater than $150\text{ N/m}$, less than $150\text{ N/m}$, or equal to $150\text{ N/m}$ ?
___ $k' > 150\text{ N/m}$ ___ $k' < 150\text{ N/m}$ ___ $k' = 150\text{ N/m}$
Justify your answer.
A uniform board of length $L$ and mass $m$ is attached to a pivot $\dfrac{L}{4}$ from the left end of the board. The left end of the board is attached to an ideal spring of spring constant $k$ that is attached to the ground. The student slowly lowers and then releases the board. The board remains at rest in the horizontal position, with the spring unstretched, as shown in Figure 1. The rotational inertia of the board about the pivot is $I$.
[Figure 1: A board of length $L$ resting horizontally, pivoted at a point $L/4$ from its left end (pivot labeled at the fulcrum). The left end of the board connects down to an "Unstretched Spring" anchored to the ground. Note: Figures not drawn to scale.]
[Figure 2: The board is shown lowered at an angle on the left side, with the spring now stretched by an amount $\Delta x$ ("Stretched Spring"), anchored to the ground; the board length $L$ is marked on the right/upper side.]
On the rectangle below, which represents the board, draw and label the forces (not components) that act on the board while the board-spring system is in equilibrium. Each force should be represented by an arrow away from the board, and should represent the location at which that force acts.
[A blank horizontal rectangle representing the board, with a dashed vertical centerline and dashed horizontal centerline, for the student to draw and label force vectors acting on the board.]
Derive an expression for the distance the spring stretches, $\Delta x$, when the board is in equilibrium. Express your answer in terms of $k$, $L$, $m$, and physical constants, as appropriate.
[Figure: A board pushed down on the right side, stretching the spring a new distance $\Delta x_2$ from the unstretched position. The board is held at a small angle $\theta_0$ with the horizontal, as shown. The student then releases the board from rest. Note: Figure not drawn to scale.]
A student pushes down the board on the right side, stretching the spring a new distance $\Delta x_2$ from the unstretched position. The board is held at a small angle $\theta_0$ with the horizontal, as shown. The student then releases the board from rest.
At time $t = 0$, the board is released. At time $t_1$, the board first crosses the horizontal. Sketch a graph of the magnitude of the angular acceleration $\alpha$ of the board as a function of time $t$ from $t = 0$ to $t = t_1$.
[A blank graph with vertical axis $\alpha$ and horizontal axis $t$, with a marked point $t_1$ on the time axis, for the student to sketch angular acceleration vs. time.]
Derive an expression for the angular acceleration $\alpha_0$ of the board immediately after the board is released. Express your answer in terms of $k$, $L$, $m$, $I$, $\Delta x_2$, $\theta_0$, and physical constants.
[Figure: The same board-spring-pivot setup, with two blocks of equal mass $m$ attached to the board equal distances from the pivot point, as shown. Note: Figure not drawn to scale.]
Two blocks of equal mass $m$ are attached to the board equal distances from the pivot point, as shown. The board is again pushed down on the right side so that the spring stretches the same distance $\Delta x_2$ as in part (c). The board is then released. The new angular acceleration $\alpha'$ when the blocks are attached compare to the angular acceleration $\alpha_0$ from part (c) ?
___ $\alpha' > \alpha_0$ ___ $\alpha' < \alpha_0$ ___ $\alpha' = \alpha_0$
Justify your answer.